A Fuzzy EWMA Attribute Control Chart to Monitor Process Mean

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by  Muhammad Zahir Khan

informatio n
Article
A Fuzzy EWMA Attribute Control Chart to Monitor
Process Mean
MuhammadZahirKhan1,MuhammadFaridKhan1,MuhammadAslam2,* ,
SeyedTaghiAkhavanNiaki3 andAbdurRazzaqueMughal1
1 DepartmentofMathematicsandStatistics,RiphahInternationalUniversity,Islamabad45210,Pakistan;
[email protected](M.Z.K.);[email protected](M.F.K.);
[email protected](A.R.M.)
2 DepartmentofStatistics,FacultyofScience,KingAbdulazizUniversity,Jeddah21551,SaudiArabia
3 DepartmentofIndustrialEngineering,SharifUniversityofTechnology,P.O.Box11155-9414AzadiAve.,
Tehran15119-43943,Iran;[email protected]
* Correspondence:[email protected];Tel.:[phone]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:30September2018;Accepted:4December2018;Published:7December2018
Abstract: Conventionalcontrolchartsareoneofthemostimportanttechniquesinstatisticalprocess
control which are used to assess the performance of processes to see whether they are in- or
out-of-control. As traditional control charts deal with crisp data, they are not suitable to study
unclear, vague, and fuzzy data. In many real-world applications, however, the data to be used
in a control charting method are not crisp since they are approximated due to environmental
uncertainties and systematic ambiguities involved in the systems under investigation. In these
situations,fuzzynumbersandlinguisticvariablesareusedtograbsuchuncertainties. Thatiswhy
the use of a fuzzy control chart, in which fuzzy data are used, is justified. As an exponentially
weightedmovingaverage(EWMA)schemeisusuallyusedtodetectsmallshifts, inthispapera
fuzzyEWMA(F-EWMA)controlchartisproposedtodetectsmallshiftsintheprocessmeanwhen
fuzzydataareavailable. Theapplicationofthenewlydevelopedfuzzycontrolchartisillustrated
usingreal-lifedata.
Keywords: crispdata;fuzzydata;fuzzycontrolcharts;EWMAcharts;fuzzyEWMAcharts
1. IntroductionandLiteratureReview
Process performance and proficiency can be enhanced by decreasing variability. This can be
attained with the assistance of statistical process control (SPC) methods. The aim of a statistical
processcontrolmethodistoachievestabilityandtoimprovetheefficiencyofaprocessbyreducing
the variability involved. Conventional control charts are one of the tools to achieve this objective.
Conventional control charts, also known as Shewhart control charts, are used to detect shifts in a
process. ThemostlyusedShewhartcontrolchartswhenavariablequalitycharacteristicismonitored
include“x-barandrange”and“x-barands(standarddeviation)”controlcharts. However,thesecharts
arenotabletodetectsmallshifts. Anotherproblemisthatthesechartsarenotindependentofeach
other. Inordertodetectsmallshiftsintheprocessmeanorintheprocessvariation,anexponentially
weightedmovingaverage(EWMA)schemeisused. Nonetheless,theconstructionmethodandthe
interpretationinvolvedinatraditionalShewhartchartaredifferentfromtheonesinanEMWAcontrol
chart. Inotherwords,theplotofaShewhartcontrolchartshowsindependentdatapointsandfollows
thelawofthesamplingdistributionofthestatistic. However,inanEWMAorinacumulativesum
(CUSUM) scheme, a statistic corresponding to a parameter is estimated directly. When a process
monitoredusinganEWMAoraCUSUMcontrolchartgoesout-of-control,thedatamovesgraduallyto
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Information2018,9,312 2of13
thenextlevel. Otherwise,thechartwillshowadiscrepancyaboutthecenterlinewithsmallvariations
whentheprocessisin-control[1]. Inaddition,theEWMAandtheCUSUMcontrolchartsareusually
usedtodetectsmallshiftsinaprocess. Yangetal.[2]proposedanewEWMAcontrolcharttomonitor
themeanofaprocesswithavariablequalitycharacteristic. Itcanbeappliedinasituationwhenthe
distributionoftheavailableprocessdataiseitherunidentifiedornon-normal. Althoughtheirchart
wasshowntohaveaverygoodperformance,theArcsine-EWMAcontrolchartdevelopedalsoby
Yangetal.[2]performedbetter. Tomonitorprocesseswithattributecharacteristics,however,thep
andthecchartsarethemostwidelyemployedschemes.
Acontrolchartconsistsofthreelines;thecenter(average)line,theupper,andthelowercontrol
limits (lines). The strength of any chart can be judged by its ability to identify the shifts and the
irregularityofaprocess. Thisstrengthisdirectlyrelatedtothechoiceofitscontrollimitswhenthe
chartisdesigned. ThisalsoappliestofuzzycontrolchartsintroducedbyRaz&Wang[3]forthefirst
time,whentheysuggestedtheuseoftwoapproaches: aprobabilisticapproachandamembership
approach. Eversince,thistypeofcharthasgainedmuchinterestintheliterature. Thecontrolcharts
arerecommendedbyKanagawa[4]forlinguistictermsobtainedbyexperts’opinionandregarded
as fuzzy data are different from the one proposed by Wang [5]. It monitors not only the process
averagebutalsotheprocessvariability. Inordertodecreasetheemergenceoffalsealarms,aswellasto
enhancethespeedtoidentifyactualerrors,El-Shal[6]usedfuzzylogictoamendSPCrules. Rowlands
&Wang[7]proposedafuzzySPCevaluationandcontrolmethodwhichcombinedthetraditional
statistical process control methodology with an intelligent system approach. Gülbay & Ruan [8]
recommendedα-cutcontrolchartstomonitorattributedatatoadjustthestiffnessoftheinspection
usingtriangularfuzzynumbers. Cheng[9]developedafuzzycontrolcharttodealwithaprocessthat
givesfuzzyoutcomes. Hedesignedtwofuzzycontrolchartstodirectlyobservethefuzzyresultsin
ordertodeterminetheprocessquality. AdirectfuzzyapproachwasintroducedbyGülbay[10]asa
substitutetofuzzycontrolcharts. Thisapproachcanbeappliedonfuzzyattributedataobtainedby
vagueevents. Tocontrolprocessmean,Faraz[11]developedafuzzycontrolchartwithawarningline
alongsideanuppercontrollimit. Erginel[12]presentedthetheoreticalstructureoffuzzyindividual
andmovingrangecontrolchartswithα-cutsusingα-levelfuzzymediantransformationtechniques.
Sentürk[13]designedfuzzy(X(cid:101) −R(cid:101))and(X(cid:101)−S(cid:101))chartswithα-cutsusingα-levelfuzzymidrange
transformationtechniques. Theyusedtriangularfuzzymembershipfunctionswithfuzzynumbers
(a,b,c). Later,Sentürk[14]proposedafuzzyregressioncontrolchart. Fuzzyvariablecontrolcharts
dealwithfuzzyvariabledata,forexample,fuzzymeasurement(length,width,height). Ontheother
hand,fuzzyattributecontrolchartsstudyqualitativedatawhichcannotbeexpressedinnumbers,
forexample,fuzzyattributes(defective,nonconformities). Whilesomefuzzyvariablecontrolcharts
include (X(cid:101) −R(cid:101)) and (X(cid:101)−S(cid:101)), fuzzy attribute control charts include (p(cid:101),c(cid:101),u(cid:101)). If the quality-related
characteristicscannotberepresentedinarithmeticform,suchascharacteristicsforappearance,softness,
color,andsoforth,thencontrolchartsforattributesareused. Productunitsareeitherclassifiedas
conformingornonconforming,dependinguponwhetherornottheymeetspecifications,orthenumber
ofnonconformities(deviationsfromspecifications)perunitiscounted. Therearefourmajortypes
of control charts. The first type, the x-chart (and related x-bar, r-, and s-charts), is a generic and
simple control chart. The x-chart is designed to be used primarily with variable data, which are
usually measurements, such as the length of an object, processing time, or the number of objects
producedperperiod. Theothertypesofcontrolchartsarethep-charts,whichareusedwithbinomial
data, the c-charts, which are used for Poisson processes, and the u-charts, generally designed for
countingdefectspersamplewhenthesamplesizevariesforeachinspection. AccordingtoKaya&
Kahraman[15]whenthevalueisgivenas“around”and”approximate”,atriangularfuzzynumber
ismostlyused,andwhenthevalueisgivenas“between”,twotrapezoidalfuzzynumbersareused.
Theexamplesofatriangularfuzzynumberandtrapezoidalfuzzynumbersareapproximatespeed
(20km)andspeedbetween(25kmto30km). Ifthevertexofatrapezoidalfuzzynumberhastakena
commonvalue,itistransformedintoatriangularfuzzynumber.
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Sentürketal.[16]introducedfuzzyucontrolchartstomonitorfuzzyattributedata. Kaya[15]
(cid:101)
designedfuzzycontrolchartsforfuzzymeasurementsoftherelatedqualitycharacteristics. Afuzzy
multivariateexponentiallyweightedmovingaverage(F-MEWMA)controlchartwassuggestedby
Alipour [17] to be implemented on a process in the food industry. Sentürk [18] suggested a fuzzy
EWMA (F-EWMA) control chart to study fuzzy univariate process data using α-cuts. The design
of this scheme is based on using triangular fuzzy numbers and the fuzzy midrange technique to
studyfuzzydatabyconvertingitintoscalarform. Theirfuturerecommendationwastoapplythe
same technique on conventional control charts to make them able to work in fuzzy environment.
Khademi[19]developedtwofuzzyapproachestodesigncontrolchartsobservingfuzzyaverages
andranges. Theseapproachesthatarebasedonfuzzymodeandfuzzybaserulesarebetterinthe
sense they do not use a defuzzification method and are more informative than the existing fuzzy
controlcharts. Theydemonstratedtheperformanceoftheirchartswiththehelpofareal-lifeexample.
Recently,Kahraman[20]introducedtwofuzzycontrolchartstomonitorfuzzyoutcomesinorderto
seewhetheraprocessisin-controlorout-of-control. Itshouldbementionedthatthecapabilityofa
processcanalsobemeasuredwiththehelpofafuzzycontrolcharttocheckwhetheraprocessisin-or
out-of-control[15].
The recommendation provided by Sentürk [18] is utilized in the current paper on another
conventional scheme proposed by Yang et al. [2] to design a new fuzzy EWMA chart that can be
usedtodetectsmallshiftsonthemeanofaprocessinafuzzyenvironment. Thismakesthenewly
designchartmoreapplicable,moreflexible,andmoreinformativeinreal-lifequalitycontrolproblems.
It gives practitioners an opportunity to study intermediate values of fuzzy parameters that is not
possiblewithcrispparametersinafuzzyenvironment. AlthoughYangetal.[2]usedapproximate
valuesintheirexample, theyaretreatedasfuzzynumbersinthecurrentwork, basedonwhicha
fuzzyformofhischartisproposed. Thisnewlydevelopedfuzzychartwillbeappliedonreal-lifedata
collectedfromthefoodindustry.
The structure of the remainder of the paper is as follows. A brief background on the fuzzy
variableandattributecontrolchartsandthefuzzytransformationapproachesisgiveninSection2.
The conventional EWMA chart proposed by Yang et al. [2] is described in Section 3 in detail.
ThefuzzyEWMAcharttomonitortheprocessmeaninafuzzyenvironmentisdesignedinSection4.
TheapplicationoftheproposedchartisinvestigatedinSection5usingcookingoilfillingdata. Finally,
theconclusionandrecommendationforfuturestudiesaregiveninSection6.
2. TheFuzzyAttributeandVariableSchemes
Fuzziness and ambiguity may have several reasons including lack of knowledge, chance,
imprecision,incompetencetoperformadequatemeasurements,andinbornfuzzinessintheprocess.
Fuzzinessinhumanisticsystemsisdealtwithinamathematicalwaywiththehelpoffuzzysets[21].
Thefuzzysettheoryprovidesanopportunitytostudyhumansubjectivityinascientificmannerwith
theassistanceoffuzzysets[22].
2.1. FuzzyTransformationApproaches
Whenfuzzydataareused,itisessentialtorepresentthefuzzysetslinkedwiththelinguisticdata
inthesamplebysomescalar(transformation)numbersforadditionalcalculations. Therearefourfuzzy
measuresofcentraltendenciesintheliteratureusedtocalculatetransformednumbers. Thesemeasures
include(1)fuzzymean,(2)fuzzymedian,(3)fuzzymode,and(4)fuzzymidrange([3,5]).
2.2. SomeFuzzyFormulae
Theformulatocomputethefuzzyaverageis[21]:
(cid:82)1 xU (X)
f = x=0 F .
avg (cid:82)1 U (X)
x=0 F
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Todeterminethefuzzymedian,thecurveunderthemembershipfunctionisdividedintotwo
halves. Asaresult,thefuzzymedianbecomes
Information 2018, 9, x FOR PEER REVIEW (cid:90) fmed 4 of 14
f = U (X)dx.
median F
a
𝑓 = (cid:1516)(cid:3033)(cid:3288)(cid:3280)(cid:3279)𝑈 (𝑋)𝑑𝑥.
Thefuzzymodeisthevalueofth(cid:3040)e(cid:3032)(cid:3031)b(cid:3036)a(cid:3028)s(cid:3041)evar(cid:3028)iablew(cid:3007)herethemembershipfunctionequals1.Thus,
The fuzzy mode is the value of the base variable where the membership function equals 1. Thus,
f = {X/U (X) =1}.
mode F
𝑓 = (cid:4668)𝑋∕𝑈 (𝑋) = 1(cid:4669).
(cid:3040)(cid:3042)(cid:3031)(cid:3032) (cid:3007)
Thefuzzymid-rangeisthemeanendpointsoftheα-cutas
The fuzzy mid-range is the mean end points of the α-cut as
1
𝑓(cid:3080)f(cid:3040)α(cid:3045)m=r =(cid:2869)(cid:2870)(2𝑎((cid:3080)a+α+𝑎(cid:3030)a)c. ).
AcAcocrcdoirndgin tgo t[o18[]1, 8t]h,etrhee raerea rneon soosliodl irdearesoansos ntso tdoedciedceid wehwichhi cthecthecnhiqnuiqe uise bisebtteetrt etor tuosue,s eb,ubt umtomsto st
resreeasrecahrecrhse arpspalipepdl tihede αt-hleevαel- lfeuvzezly fmuizdzryanmgeid terachnngiequteec ihnn thiqeuire winorkths ediurew too ritkss sidmupelet oforitms osifm foprmleufloar. m
offormula.
2.3. Triangular Fuzzy Number
2.3. TriangularFuzzyNumber
A triangular fuzzy number 𝐴(cid:4634) is a fuzzy number with its membership function defined by three
numberAs t𝑎rian<g𝑎ula<rf𝑎uz zwyhneurme btheer Ab(cid:101)aissea offu zthzye ntruiamnbgeler wisi tthheit sinmteermvable r(cid:4670)s𝑎hi,p𝑎f(cid:4671)u nacntido nthdee fivneretdexb yist hart ee
(cid:2869) (cid:2870) (cid:2871) (cid:2869) (cid:2871)
𝑋 =nu𝑎m(cid:2870).b ersa1 < a2 < a3wherethebaseofthetriangleistheinterval[a1,a3]andthevertexisatX = a2.
ExEamxapmlep.l Ien. Itnhet hfoellfoowlloinwgi nFgigFuigreu r1e, 1th,etrhee rise itsritarniagnuglaurl anrunmubmebr edredfienfiende bdyb ythtrheree neunmumbebresr (sa(1a =1 =2, 2a,2 a=2 3=, 3a3, a=3 4=), 4),
whwerhee rtehet hbeasbea soef othfet hteritarniagnleg ilse tishet hinetienrtvearvl a(cid:4670)l2[,24,(cid:4671)4 ]anand dthteh evveretretxex isi sata t𝑋X==33. .

Figure1.TriangularFuzzyNumber.
Figure 1. Triangular Fuzzy Number.
The basic forms of the variable and the attribute fuzzy control charts were given in [20].
ThTehaet tbriabsuict efocromnstr oofl tchhea rvtsaraiarbelep -acnhda rtth,en pa-ttcrhibaurtt,e cf-uchzzayrt ,coanntdroul -cchhaarrtts, wwehriele gtihveenv ainr ia[2b0l]e. cThhaer ts
(cid:101) (cid:101) (cid:101) (cid:101)
attirnibcluutdee c(oX(cid:101)nt−rolR(cid:101) c)haanrdts (aX(cid:101)re− 𝑝(cid:3556)S(cid:101)-c)hsacrhte, m𝑛𝑝e(cid:3556)-sc.hFaurzt,z y𝑐̃-fcohramrst, oafntdh e𝑢(cid:3556)se-cchhaarrtt, swmhailkee tthhee mvarmiaobrleefl cehxairbtlse ,insecnlusditeiv e,
(cid:3435)𝑋(cid:3364)(cid:3560)a−nd𝑅(cid:3561) i(cid:3439)n faonrdm a(cid:3435)t𝑋(cid:3560)iv−e.𝑆(cid:4634)A(cid:3439) sfuchzezmyevsa.r Fiaubzlzeyq fuoarlmitsy ocfh tahreascet ecrhisatritcs imnamkes athmepmle ms,oeraec hflewxiibthle,t hseenssiizteivoef, annwd as
infroerpmreasteivnete. dAu sfuinzgzya tvraiarniagbulela rqfuuazliztyy ncuhmarbaecrtedriesntioct eidn b𝑚y (cid:102)Xsaim=p(lXesia, ,eXaicbh,X wic)it,hw thheer esiiz=e 1o,f2 ,𝑛3 ,w..a.s, m.
repInreasdendtietido nu,sienagc ha tsraiamnpgluelainr faufzuzzyz nyuamttbreibru dteencohtaedrt ,bfyo 𝑋r(cid:3561)(cid:3114)e=xa(m𝑋p(cid:3036)(cid:3028)l,e𝑋,(cid:3036)t(cid:3029)h,𝑋e(cid:3036)n(cid:3030))p(cid:101), wcohnetrreo l𝑖c=ha1r,t2d,3e,v…e,lo𝑚p. eIdn to
adsdtuitdioynt,h eeacnhu smabmeprloef inno an fcuoznzfyo ramttirnibguutne icthsainrta, ffourz ezxyaemnpvliero, nthme e𝑛n𝑝t(cid:3556), wcoanstprorel scehnatretd dbeyvealotrpiaendg tuol satrufduyz zy
the number of nonconforming units in a fuzzy environment, was presented by a triangular fuzzy
number (𝑑 ,𝑑 ,𝑑 ) with the mean of (cid:3435)𝑛𝑝̅ 𝑛𝑝̅ 𝑛𝑝̅ (cid:3439), where 𝑛𝑝̅ =∑(cid:3288)(cid:3284)(cid:3128)(cid:3117)(cid:3031)(cid:3284)(cid:3276),𝑛𝑝̅ =∑(cid:3288)(cid:3284)(cid:3128)(cid:3117)(cid:3031)(cid:3284)(cid:3277),and 𝑛𝑝̅ =
(cid:3036)(cid:3028) (cid:3036)(cid:3029) (cid:3036)(cid:3030) (cid:3028), (cid:3029), (cid:3030) (cid:3028) (cid:3040) (cid:3029) (cid:3040) (cid:3030)
∑(cid:3288)(cid:3284)(cid:3128)(cid:3117)(cid:3031)(cid:3284)(cid:3278). Hence, the control limits of the fuzzy 𝑛𝑝(cid:3556) chart are [20]:
(cid:3040)
𝑈(cid:3563)𝐶𝐿=(cid:4672)𝑛𝑝̅ +3(cid:3493)𝑛𝑝̅ (1−𝑝̅ ),𝑛𝑝̅ +3(cid:3493)𝑛𝑝̅ (1−𝑝̅ ),𝑛𝑝̅
(cid:3028) (cid:3028) (cid:3028) (cid:3029) (cid:3029) (cid:3029) (cid:3030)
(1)
+3(cid:3493)𝑛𝑝̅ (1−𝑝̅ )(cid:4673)
(cid:3030) (cid:3030)
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number(d ,d ,d )withthemeanof(np np np ),wherenp = ∑im=1dia,np = ∑im=1dib,andnp =
ia ib ic a, b, c a m b m c
∑im=1dic. Hence,thecontrollimitsofthefuzzynpchartare[20]:
m (cid:101)
(cid:113) (cid:113) (cid:113)
(cid:93)
UCL = (np +3 np (1−p ),np +3 np (1−p ),np +3 np (1−p )) (1)
a a a b b b c c c
C(cid:102)L = (np ,np ,np ) (2)
a b c
(cid:113) (cid:113) (cid:113)
L(cid:103)CL = (np −3 np (1−p ),np −3 np (1−p ),np −3 np (1−p )). (3)
a a a b b b c c c
3. TheEWMAControlCharttoMonitorProcessMean
AnewversionoftheconventionalEWMAcontrolchart,initiallypresentedbyMontgomery[23],
wasproposedbyYangetal.[2]tomonitorsmallshiftsinaprocessmean. Thedetailsofthischartare
givenasfollows. ArandomsampleofsizenincludingtherandomvariablesX X X ...,X istaken
1, 2, 3, n
fromaprocesswithameanµinsubgroupi.Let
Y = X −µ; j =1,2,3,..., n, i =1,2,3,...,m. (4)
j ij
Inasubgroup,defineaBernoullirandomvariableas
(cid:40)
1; Y >0
I = j . (5)
j
0; Otherwise
Let S be the total number of Y > 0. Then, S = ∑n I would follow a binomial distribution
j j=1 j
(cid:8) (cid:9)
withtheparameters(n,p)foranin-controlprocesswith p = P Y >0 astheprobabilityofhaving
j
= =
a positive difference (X −X) > 0, in which X is the estimate of the process mean µ. Although
ij
the S statistic in Yang et al.’s EWMA chart [2] follows a binomial distribution, the difference with
theconventionalnpchartisthatthebinomialvariableisnotthecountofnon-conformingunitsin
thesample,ratherthenumberof X valuesinasamplethatareabovethein-controlprocessmean.
i
Thischartisdesignedtodetectsmallshiftsrapidlyandeffectively.
TheEWMAstatisticsinYangetal.’sEWMAchart[2]isdefinedas
EWMA = λS +(1−λ)EWMA ; 0< λ ≤1 (6)
Si i Si
whereS representsthevalueofSintheithsubgroup.TheinitialvalueofthisstatisticisEWMA = np.
i Si
AsthemeanofthestatisticisE(EWMA ) = npandthevarianceisVar(EWMA ) = np(1−p)with
Si Si
thelimitingvalueofVar(EWMA ) = λ [np(1−p)],thendependingonthenumberofthesubgroup
Si 2−λ
inwhichasampleisdrawn,theEWMAchartproposedbyYangetal.[2]isdesignedbythecontrol
limitsinEquations(7)–(9)and(10)–(12)foralargeandasmallnumberofsubgroup,respectively.
Whenthesubgroupnumberislarge:
(cid:115)
(cid:18) (cid:19)
λ
UCL = np+k [np(1−p)] (7)
EWMAS 2−λ
CL = np (8)
EWMAS
(cid:115)
(cid:18) (cid:19)
λ
LCL = np−k [np(1−p)] . (9)
EWMAS 2−λ
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Whenthesubgroupnumberissmall:
(cid:115)
(cid:18) (cid:19)
λ
UCL = np+k [np(1−p)] [1−(1−λ)2t] (10)
EWMAS 2−λ
CL = np (11)
EWMAS
(cid:115)
(cid:18) (cid:19)
λ
LCL = np−k [np(1−p)] [1−(1−λ)2t]. (12)
EWMAS 2−λ
ThisversionoftheEWMAchartisusedinthecurrentworktodesignafuzzyEWMAchartin
Section4. However,beforeproposingthechart,abriefbackgroundisprovidedinthenextsectionto
describeanexistingfuzzyEWMAchart.
TheExistingFuzzyEMWAControlChart
In the fuzzy EWMA scheme proposed by Sentürk [18], z is the tth exponentially weighted
t
movingaverage,x indicatesthesamplemean,0< λ <1isarealconstant,Xistheoverallsample
t
=
mean,misthenumberofsamples,t =1,2,3...misthesamplenumber,andz = X. Assumingx s
0 i
tobeindependentrandomvariableswithaknownvariance σ2, thevarianceof z becomes σ2 =
n t zt
σ2( λ )[1−(1−λ)2t], where n. is the samplesize. Then, when t is small the control limits in the
n 2−λ
traditionalEWMAcontrolchartareasfollows:
(cid:115)
= σ2(cid:18) λ (cid:19)
UCL = X+3 [1−(1−λ)2t] (13)
EWMA n 2−λ
=
CL = X (14)
EWMA
(cid:115)
= σ2(cid:18) λ (cid:19)
LCL = X−3 [1−(1−λ)2t]. (15)
EWMA n 2−λ
However,astincreasesσ2 tendstoalimitingvalueσ2 = σ2( λ ). Assuch,thecontrollimits
zt zt n 2−λ
inthetraditionalEWMAcontrolchartbecome
(cid:115)
= σ2(cid:18) λ (cid:19)
UCL = X+3 (16)
EWMA n 2−λ
=
CL = X (17)
EWMA
(cid:115)
= σ2(cid:18) λ (cid:19)
LCL = X+3 . (18)
EWMA n 2−λ
Ontheotherhand,thecontrollimitsofthefuzzyEWMAschemeproposedby[18]whenσis
knownandtissmallare
(cid:115)
= = = 3 (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)+ √n(σa,σb,σc) 2−λ [1−(1−λ)2t] (19)
= = =
C(cid:101)LEWMA = (Xa,Xb,Xc) (20)
(cid:115)
= = = 3 (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)− √n(σa,σb,σc) 2−λ [1−(1−λ)2t]. (21)
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However,whentislarge,thecontrollimitsbecome
(cid:115)
= = = 3 (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)+ √n(σa,σb,σc) 2−λ (22)
= = =
C(cid:101)LEWMA = (Xa,Xb,Xc) (23)
(cid:115)
= = = 3 (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)− √n(σa,σb,σc) 2−λ . (24)
Incaseσisunknownandtissmall,thecontrollimitsare
(cid:115)
= = = (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)+3(Ra,Rb,Rc) 2−λ [1−(1−λ)2t] (25)
= = =
C(cid:101)LEWMA = (Xa,Xb,Xc) (26)
(cid:115)
= = = (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)−3(Ra,Rb,Rc) 2−λ [1−(1−λ)2t]. (27)
Finally,whenσisunknownandtislargethecontrollimitsare
(cid:115)
= = = (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)+3(Ra,Rb,Rc) 2−λ (28)
= = =
C(cid:101)LEWMA = (Xa,Xb,Xc) (29)
(cid:115)
= = = (cid:18) λ (cid:19)
UC(cid:101)LEWMA = (Xa,Xb,Xc)−3(Ra,Rb,Rc) 2−λ . (30)
NoteinEquations(25)–(30)thatR , R , andR areranges.
a b c
4. TheProposedFEWMASchemeBasedonthenpchart
Based on the background provided in Section 3, the control limits of the new fuzzy EWMA
schemetodetectsmallshiftsintheprocessmeanwhenfuzzyprocessdataareavailableareproposed
asfollows. SimilartothelimitsshownintheexistingfuzzyEWMAchart, assuminganunknown
standarddeviationoftheprocess,thecasethatmostlyhappensinpractice,thelimitsareprovided
dependingonwhethertissmallorlarge.
4.1. σIsUnknownandtIsSmall
Thecontrollimits,inthiscase,areproposedas
U(cid:93)CLEWMAS =(npa,npb,npc)+K(cid:114)(cid:16)2−λλ(cid:17)[1−(1−λ)2t][npa(1−pa)],[npb(1−pb)],[npc(1−pc)] (31)
C(cid:102)LEWMAS = (npa,npb,npc) (32)
(cid:114)(cid:16) (cid:17)
L(cid:103)CLEWMAS =(npa,npb,npc)−K 2−λλ [1−(1−λ)2t][npa(1−pa)],[npb(1−pb)],[npc(1−pc)] (33)
Information2018,9,312 8of13
whichcanbewritteninthefollowingforms:
(cid:93) (cid:18) (cid:114)(cid:16) (cid:17)
UCL = np +K λ [1−(1−λ)2t][np (1−p )], np
EWMAS a 2−λ a a b
(cid:114)
(cid:16) (cid:17)
+K λ [1−(1−λ)2t][np (1−p )], np (34)
2−λ b b c
(cid:114)(cid:16) (cid:17) (cid:19)
+K λ [1−(1−λ)2t][np (1−p )]
2−λ c c
C(cid:102)LEWMAS = (npa,npb,npc) (35)
(cid:18) (cid:114)(cid:16) (cid:17)
L(cid:103)CLEWMAS = npa−K 2−λλ [1−(1−λ)2t][npa(1−pa)], npb−
(cid:114)
(cid:16) (cid:17)
K λ [1−(1−λ)2t][np (1−p )], np − (36)
2−λ b b c
(cid:114)(cid:16) (cid:17) (cid:19)
K λ [1−(1−λ)2t][np (1−p )]
2−λ c c
In Equations (34) and (36), the parameter K is chosen such that the chart exhibits a desired
in-controlaveragerunlength.
4.2. σIsUnknownandtIsLarge
Thecontrollimitsinthiscaseare
(cid:115)
(cid:18) (cid:19)
(cid:93) λ
UCL = (np ,np np )+K [np (1−p )],[np (1−p )],[np (1−p )]) (37)
EWMAS a b, c 2−λ a a b b c c
C(cid:102)LEWMAS = (npa,npb,npc) (38)
(cid:115)
(cid:18) (cid:19)
λ
L(cid:103)CLEWMAS = (npa,npb,npc)−K 2−λ [npa(1−pa)],[npb(1−pb)],[npc(1−pc)] (39)
whichcanberewrittenas
(cid:93) (cid:18) (cid:114)(cid:16) (cid:17)
UCL = np +K λ [np (1−p )], np
EWMAS a 2−λ a a b
(cid:114)
(cid:16) (cid:17)
+K λ [np (1−p )], np (40)
2−λ b b c
(cid:114)(cid:16) (cid:17) (cid:19)
+K λ [np (1−p )]
2−λ c c
C(cid:102)LEWMAS = (npa,npb,npc) (41)
(cid:16) (cid:113)
L(cid:103)CLEWMAS = npa−K (2−λλ)[npa(1−pa)],npb
(cid:113)
−K ( λ )[np (1−p )], np (42)
2−λ b b c
(cid:113) (cid:17)
−K ( λ )[np (1−p )] .
2−λ c c
Then,dependingonthefuzzymeasuresofcentraltendenciesusedforthefuzzytransformation
approachdiscussedinSection2.2,thefollowinglimitsareobtained.
4.3. TheControlLimitsintheα-CutsFuzzyEWMAControlChart
Thecontrollimitsinthisfuzzytransformationapproachare
Information2018,9,312 9of13
(cid:94) (cid:16) (cid:113)
UCLα = np α+K λ (np α(1−p α)), np α
EWMAS a 2−λ a a b
(cid:113)
+K λ (np α(1−p α)), np α (43)
2−λ b b c
(cid:113) (cid:17)
+K λ (np α(1−np α))
2−λ c c
C(cid:103)LαEWMAS = (npaα,npbα,npcα) (44)
(cid:93) (cid:16) (cid:113)
LCLα = np α+K λ (np α(1−p α)),np α
EWMAS a 2−λ a a b
(cid:113)
+K λ (np α(1−p α)),np α (45)
2−λ b b c
(cid:113) (cid:17)
+K λ (np α(1−np α)) .
2−λ c c
4.4. TheControlLimitsintheα-CutFuzzyMedianEWMAScheme
ThecontrollimitsoftheproposedfuzzyEWMAschemewhenthistypeoffuzzytransformation
approachisusedareshowninEquations(46)–(48).
U(cid:94)CLαmed−EWMAS = 13(npaα+npbα+npcα)+13K(cid:113)2−λλ(npaα(1−paα))+(npbα(1−pbα))+(npcα(1−npcα)) (46)
1
C(cid:103)Lαmed−EWMAS = 3(npaα+npbα+npcα) (47)
L(cid:93)CLαmed−EWMAS = 13(npaα+npbα+npcα)−13K(cid:113)2−λλ(npaα(1−paα))+(npbα(1−pbα))+(npcα(1−npcα)) (48)
5. TheApplication
In this section, the application of the proposed fuzzy EWMA chart is demonstrated using
fuzzy observations obtained from a cooking oil filling process involved in the food industry in
Pakistan. Inthisexample,fuzzydataarecollectedfromtheprocess,wherefuzzyobservationsare
treatedastriangularfuzzynumbers(x ,x ,x )ineachsampletakenatdifferenttimes. Inaddition,
ai bi ci
theproportionofthevaluesaboveagiventhreshold(zero)willbetreatedasatriangularfuzzynumber
(p ,p ,p ). Thedataconsistof10samples,eachwith10fuzzyobservations,andtheproportionis
ai bi ci
obtainedfromeachsamplebasedonthegiventhreshold.
In this application, the chart parameters used are K = 2.58, λ = 0.2, n = 10, p = 0.35, p = 0.48,
a b
α=0.65,andp =0.64. Here,theparametersettingK=2.58,λ=0.2,andP=0.65isobtainedusinga
c
simulationapproachdescribedinthenextsection. Inaddition, p = 0.35showstheproportionof“a”
a
aboveitsgivenmean,p =0.48impliestheproportionabovethemeanof“b”,andp =0.64indicates
b c
theproportionabovethemeanof“c”. Thisisdoneduetothefactthateachvalueisbeingusedasa
triangularfuzzynumber. Thecontrollimitsarecalculatedas
(cid:93)
UCL == (4.75, 6.117.66)
EWMAS
C(cid:102)LEWMAS == (3.5, 4.8, 6.4)
L(cid:103)CLEWMAS == (2.22,4.46,5.13).
Then,thelimitsoftheα-cutsfuzzyEWMAcontrolchartareobtainedbasedonEquations(43)–(45)as:
(cid:94)
UCLα == (5.69,6.15,6.7)
EWMAS
(cid:93)
LCLα == (2.99, 3.44, 4.0).
EWMAS
However, when the α-cut fuzzy median EWMA scheme is used, the control limits based on
Equations(46)–(48)areobtainedas
Information2018,9,312 10of13
(cid:94)
UCLαmed−EWMAS =5.62
C(cid:103)Lαmed−EWMAS =4.84
(cid:94)
LCLαmed−EWMAS =4.05.
Whentheα-cutfuzzymedianEWMAchartisemployedonthefirstsamplewehave
np α = np +α(np −np ) (49)
a a,1 a,1 b,1 a,1
npα = np +α(np −np ) (50)
c,1 c,1 b,1 c,1
1
Sαmed−EWMAsj = (npαa,j+npb,j+ npαc,j); j =1,2,... . (51)
3
Then,atα=0.65wehave
np 0.65 = 4.65
a a,1
np =5
b,1
np0.65 =5.35.
c,1
Asaresult,thestatisticinthefirstsamplebecomes
Sαmed−EWMAs1 ==5
Now,sincethestatisticfallswithinthecontrollimit,i.e.,4.05<5<5.61,theprocessisin-control
basedonthefirstsampledrawn.
Theproportionsofthefuzzyobservationsabovetheprocessmeanarefirstcalculatedforeach
sampleinTable1. ThedetailedcalculationusingtheRstatisticalsoftwareisgiveninAppendixA.
Then,thecontrollimitsoftheproposedfuzzyexponentiallyweightedmovingaveragecontrolchart
usingtheα-cutmedianapproacharecomputed. Theresultsshowthattheprocessisin-control.
Table1.Monitoringthefuzzyoilpackagingprocessusingtheproposedfuzzyexponentiallyweighted
movingaverage(EWMA)scheme.
Sample FuzzyProportion α-CutsfortheProportion FuzzyEWMAs Fuzzy(EWMA-med) 4.054<SαEMWAsα<5.62
1 (4,5,6) (4.65,5,5.35) (3.6,4.84,6.32) 4.92 In-control
2 (3,4,5) (3.65,4,4.35) (3.8,4.8,5.8) 4.80 In-control
3 (5,6,7) (5.65,6,6.35) (3.4,4.4,5.4) 4.40 In-control
4 (2,3,4) (2.65,3,3.35) (5.05,5.4,5.75) 5.40 In-control
5 (6,5,7) (6.65,6,6.35) (4.80,6.3,6.46) 4.60 Incontrol
6 (3,4,5) (3.65,4,4.35) (4.6,5.6,6.6) 4.93 In-control
7 (4,6,7) (4.65,3,2,3.25) (3.40,4.6,4.7) 4.63 In-control
8 (3,4,5) (3.65,4,4.35) (4.6,5.6,6.6) 4.93 In-control
9 (4,5,6) (4.65,5,5.35) (3.2,4.2,5.2) 4.20 In-control
10 (6,7,8) (6.65,7,7.35) (4.4,5.4,6.4) 5.40 In-control
TheSimulation
InthissectionwesimulatedthedatausingMATLABcodetocalculateFuzzyaveragerunlength
(ARL1s) when the process is not in-control shown in Table 2. Similarly, conventional ARL1s are
presentedinTable3. ComparisonshowsthatthefuzzyARL1sarelessthantheconventionalARL1s.
Thismeansthatthefuzzycontrolschemeisquickerinindicatingsmallshiftsintheprocessascompared
totheconventionalchart. Thisisthestrengthoftheproposedfuzzycontrolchartcomparedtothe
conventionalchart.
ThefollowingalgorithmwasusedtofindthecontrolchartcoefficientsandtheARLs.
1. Generateabinomialrandomvariable.